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 Uspekhi Mat. Nauk, 2014, Volume 69, Issue 4(418), Pages 103–168 (Mi umn9568)

Interpolation functions and the Lions–Peetre interpolation construction

V. I. Ovchinnikov

Voronezh State University

Abstract: The generalization of the Lions–Peetre interpolation method of means considered in the present survey is less general than the generalizations known since the 1970s. However, our level of generalization is sufficient to encompass spaces that are most natural from the point of view of applications, like the Lorentz spaces, Orlicz spaces, and their analogues. The spaces $\varphi(X_0,X_1)_{p_0,p_1}$ considered here have three parameters: two positive numerical parameters $p_0$ and $p_1$ of equal standing, and a function parameter $\varphi$. For $p_0\ne p_1$ these spaces can be regarded as analogues of Orlicz spaces under the real interpolation method. Embedding criteria are established for the family of spaces $\varphi(X_0,X_1)_{p_0,p_1}$, together with optimal interpolation theorems that refine all the known interpolation theorems for operators acting on couples of weighted spaces $L_p$ and that extend these theorems beyond scales of spaces. The main specific feature is that the function parameter $\varphi$ can be an arbitrary natural functional parameter in the interpolation.
Bibliography: 43 titles.

Keywords: interpolation spaces, interpolation functors with function parameters, interpolation orbits, orbits with respect to von Neumann–Schatten operators, optimal interpolation theorems, embedding theorems for Orlicz–Sobolev spaces.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00378 This research was supported by the Russian Foundation for Basic Research (grant no. 13-01-00378).

DOI: https://doi.org/10.4213/rm9568

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English version:
Russian Mathematical Surveys, 2014, 69:4, 681–741

Bibliographic databases:

UDC: 517.982
MSC: Primary 46B70; Secondary 46M35, 47A57

Citation: V. I. Ovchinnikov, “Interpolation functions and the Lions–Peetre interpolation construction”, Uspekhi Mat. Nauk, 69:4(418) (2014), 103–168; Russian Math. Surveys, 69:4 (2014), 681–741

Citation in format AMSBIB
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This publication is cited in the following articles:
1. L. Kussainova, A. Ospanova, “Interpolation theorems for weighted Sobolev spaces”, World Congress on Engineering, WCE 2015, V. I, Lecture Notes in Engineering and Computer Science, eds. Ao S., Gelman L., Hukins D., Hunter A., Korsunsky A., Int. Assoc. Engin., 2015, 25–28
2. V. I. Dmitriev, “On one transformation of parameter-spaces of real interpolation method”, Russian Math. (Iz. VUZ), 60:12 (2016), 36–42
3. Gogatishvili A. Neves J.S., “Weighted Norm Inequalities For Positive Operators Restricted on the Cone of Lambda-Quasiconcave Functions”, Proc. R. Soc. Edinb. Sect. A-Math., 150:1 (2020), 17–39
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